| import Mathlib |
| import OAI.Geometry.Cannon.Modulus.Weights |
|
|
| namespace OAI |
|
|
| section |
| section |
| section |
| noncomputable section |
| section |
| section |
| section |
| section |
| section |
| open Set Topology |
| open Set Metric Topology |
| open Set Topology Filter Metric unitInterval |
| open scoped InnerProductSpace |
|
|
| namespace CannonCells |
|
|
| open Set |
|
|
| variable {X ι : Type*} [Fintype ι] |
|
|
| noncomputable def cost (C : ι → Set X) (w : ι → ℝ) (A : Set X) : ℝ := |
| ∑ i ∈ indices C A, w i |
|
|
| lemma cost_nonneg (C : ι → Set X) {w : ι → ℝ} (hw : ∀ i, 0 ≤ w i) (A : Set X) : |
| 0 ≤ cost C w A := Finset.sum_nonneg (fun i _ => hw i) |
|
|
| lemma indices_mono (C : ι → Set X) {A B : Set X} (h : A ⊆ B) : indices C A ⊆ indices C B := by |
| intro i hi |
| obtain ⟨x,hx,hxi⟩ := (mem_indices C A i).mp hi |
| exact (mem_indices C B i).mpr ⟨x,h hx,hxi⟩ |
|
|
| lemma cost_mono (C : ι → Set X) {w : ι → ℝ} (hw : ∀ i, 0 ≤ w i) |
| {A B : Set X} (h : A ⊆ B) : cost C w A ≤ cost C w B := by |
| classical |
| exact Finset.sum_le_sum_of_subset_of_nonneg (indices_mono C h) (fun i _ _ => hw i) |
|
|
| lemma indices_union [DecidableEq ι] (C : ι → Set X) (A B : Set X) : |
| indices C (A ∪ B) = indices C A ∪ indices C B := by |
| classical |
| ext i |
| simp only [mem_indices,Finset.mem_union,union_inter_distrib_right,union_nonempty] |
|
|
| lemma cost_union_le (C : ι → Set X) {w : ι → ℝ} (hw : ∀ i, 0 ≤ w i) (A B : Set X) : |
| cost C w (A ∪ B) ≤ cost C w A + cost C w B := by |
| classical |
| unfold cost |
| rw [indices_union] |
| have h := Finset.sum_union_inter (s₁ := indices C A) (s₂ := indices C B) (f := w) |
| have hn : 0 ≤ ∑ i ∈ indices C A ∩ indices C B, w i := |
| Finset.sum_nonneg (fun i _ => hw i) |
| linarith |
|
|
| end CannonCells |
|
|
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
|
|
| end OAI |
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|
